[Hanzo]

I know, I know you are sick and tired of this but...
Right now my pals and I are runnning SR5 campaing. We are solving all questions together, but I've faced 2 issues. I've browesed through half of the forum but still have no anwser. So...
1) AR this is a combitation of the meat world and the Matrix. You can use your legs to go anywhere and interract with any object both physical and Matrix. But can you enter Hosts in AR? Do you need first to enter, say library, in the meet world and then enter the library's host? Or you can, seeting at home, use Matrix Perception to find library's host and enter it buy request? (My guess that the second option is available only for VR)
2) Do I understand correct that any site that provides services (i.e. cleaning servise site) is a host with Rating 1-2, and when you interract with an advertising agent or link you get invitation mark and enter a host?

[Hobbes]

Yes you can enter Hosts in AR.  Whatever you're viewing the Matrix through is now the Host instead of the local Matrix Icons.  The CRB suggests a -2 penalty to meat space tests when you're focusing on the Matrix.  I personally suggest simply sitting down is a good idea.

Yes you can enter a host from anywhere in the world.  One of the upsides of Hosts.

And yes frequently many commercial hosts are offering marks up to anyone who wants to come in and shop/book/browse/whatever.  Ratings vary tremendously.  But simple transactions are easily done via AR menus without entering a Host.  Vending machines, restaurants, retail shopping, and such are all handled in AR without a Host, ez-pz.  Very likely a Host on the Back End "doing stuff" that game mechanics don't detail because B2B transactions are boring and complex and should just happen transparently. 

[Xenon]

1) AR this is a combitation of the meat world and the Matrix.

AR in its most basic form is when you look at and touch the screen of your commlink (like people did back in 2020). In earlier editions of Shadowrun you would normally interact with your commlink and the matrix as a whole by using AR glasses, AR glove and earbuds. In SR5, however, most people interact with their commlink directly with their brains by wearing trodes or implanting a datajack (this also give them access to "super AR" which also include smell and taste).

People are basically always considered to be connected to the matrix through their commlinks.
Same as people back in 2020 would always be considered to be connected to the internet through their smartphones.

If you are connected to the matrix (which is basically always) then you are also in AR mode.
- You are not experience AR if you are using VR or Astral Perception/Projection.
- You are also not experiencing AR if your commlink is destroyed or placed inside a Faraday's cage or is turned off etc.

You can use your legs to go anywhere and interract with any object both physical and Matrix. But can you enter Hosts in AR?

Sure.

Same as people back in 2020 could walk around at downtown Seattle and at the same time browse a site in Japan on their smartphone.

Do you need first to enter, say library, in the meet world and then enter the library's host?

Nope.

You can sit at a cafe, drinking your soycof, and at the same time using your commlink to access the library's host. The only requirement is that you have a mark on the host (if it is a public library then the library's host will probably invite you to place your mark on it).

Or you can, seeting at home, use Matrix Perception to find library's host and enter it buy request?

You don't even need to use Matrix Perception.
The distance to all hosts are considered to be zero and they are immediately obvious.
No test is required.
(Unless the host is running silent, you didn't already spot it and you don't have access on the host already)

The CRB suggests a -2 penalty to meat space tests when you're focusing on the Matrix. 

SR5 p. 229 Augmented Reality
If your attention is really focused on your AR display and not your surroundings, your gamemaster may impose a –2 dice pool penalty on any Perception tests you make to notice things going on around you in physical space.

[BeCareful]

Similarly, you can move your Persona independently of yourself in AR. You don't need VR to do that. It's part of why you can't tell where someone is in meatspace by just looking at their Persona. (You need to do the "Trace Icon" Matrix Action for that, which you can do automatically for any device you own)